Optimization and Optimal Control

The last but one chapter in this book is devoted to batch scheduling and planning. These problems are part of optimization problems and hence an industrial case study illustrate how to formulate such problems and solve it. Batch process simulation is the last chapter, mostly devoted to batch process simulation software and illustrative case studies. 

This chapter provided the introduction to various unit operations described in this book. Introduction to solution techniques involved in solving differential equations, optimization, and optimal control problems commonly encountered in batch processing is also presented. 

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Oh, P. P., Ray, A. K. and Rangaiah, G. P. (2002a). Optimal design and operation of an industrial hydrogen plant for multiple objectives, in Recent developments in optimization and optimal control in chemical engineering edited by R. Luus, Research Signpost, Trivandrum, India, pp. 289-306. 

Ktipper, A. and Engell, S. (2011) Optimization of simulated moving bed processes, in Constrained Optimization and Optimal Control of Partial Differential Equations (eds G. Leugeringet oL), Birkhauser, Basel, pp. 559—582. 

Uncertainty analysis was mentioned many times at the workshop, and still to come are multiscale systems, fully stochastic systems, and so on. And then we will move toward design optimization and optimal control. There is a need for more and better software tools in this area. Even further in the future will be computational design of experiments, with the challenge of learning the extent to which one can learn something from incomplete information. Where should the experiment be done Where does the most predictive power exist in experiment space and physical space Right now, these questions are commonly answered by intuition, and some experimentalists are tremendonsly good at it, but computations will be able to help. Some examples are shown in Box 3. 

Batch Distillation Chapter 4 is devoted to batch distillation. This is one of the most important and one of the most studied unit operations in batch industries. Separation is based on vapor-liquid equilibria. There are a number of configurations possible in conventional batch colmnn, namely, the constant reflux mode, the variable reflux mode, and the optimal reflux mode. There are a number of new configurations that have emerged in the literature for batch distillation. This chapter describes aU these operating modes and configurations. Various levels of models are available for different analysis. Different numerical integration techniques are needed to solve equations of these different models. Optimization and optimal control are well studied for this unit operation. 

Batch Crystallization Chapter 10 is devoted to batch crystallization where a phase diagram is used to find the supersaturation at which point material crystallizes. This is again one of the most studied batch operations. Similar to batch distillation, various modeling techniques are used to describe the operation of batch crystallizer, and optimization and optimal control problems are well studied. 

This temperature dependency is exploited in optimal control problems of batch reactor where optimal temperature profile is obtained by either maximizing conversion, yield, profit, or minimizing batch time for the reaction. One of the earliest works on optimal control of batch reactor was presented by Denbigh[25] where he maximized the yield. The review paper by Srinivasan et al.[26] describes various optimization and optimal control problems in batch processing and provides examples of semi-batch and fed-batch reactor optimal control. 

There are numbers of different methods to solve an optimal control problem as discussed in the chapter on optimization and optimal control. Benavides and Diwekar (2011, 2013)[32, 34] used the maximum principle to solve the maximum concentration problem where the batch time was fixed at 100 minutes. Figure 3.10 shows the concentration profile for the base case versus the profile obtained using optimal temperature profile shown in Figure 3.11. It can be seen that the optimal concentration for methyl ester is found to be 0.7944 mol/L, while at constant temperature, the maximum concentration is 0.7324 mol/L (8.46% gain). Alternatively, if we fix the concentration at 0.7324 mol/L, the reaction time needed would be 69.5% less than it was at the beginning. 

Despite the advances in the thermodynamics for predicting azeotropic mixture, feasible distillation boundaries, and sequence of cuts, the azeotropic batch distillation system is still incipient in terms of design, optimization, and optimal control. 

Numerical optimization plays an important role in batch processing. Whether to find maximum yield in the reactor, or maximum distillation in batch distillation, or optimal schedule for batch processing, optimization and optimal control methods are extensively used. In general, the problems in batch and bio processing are large scale problems where analytical solntions are difficult. Hence nnmerical optimization methods are necessary. 

Urmila M. Diwekar and K. P. Madhavan. Batch-dist A comprehensive package for simulation, design, optimization and optimal control of multicomponent, multifraction batch distillation columns. Comp. Chem. Eng., 15(12) 833-842, 1991. 

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